Beating the standard quantum limit:: phase super-sensitivity of N-photon interferometers

Quantum metrology promises greater sensitivity for optical phase measurements than could ever be achieved classically. Here, we present a theory of the phase sensitivity for the general case where the detection probability is given by an N photon interference fringe. We find that the phase sensitivity has a complicated dependence on both the intrinsic efficiency of detection eta and the interference fringe visibility V. Most importantly, the phase that gives maximum phase sensitivity is in general not the same as the phase at which the slope of the interference fringe is a maximum, as has previously been assumed. We determine the parameter range where quantum enhanced sensitivity can be achieved. In order to illustrate these theoretical results, we perform a four-photon experiment with eta = 3/4 and V = 82 +/- 6% (an extension of our previous work (Nagata et al 2007 Science 316 726)) and find a phase sensitivity 1.3 times greater than the standard quantum limit at a phase different to that which gives maximum slope of the interference fringe.